The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n)

2008/11/06 by Mark Colarusso, Colarusso, Mark
Mathematics · #14L30 #14R20 #37K10 #53D17 #FOS: Mathematics #Group Theory (math.GR) #Symplectic Geometry (math.SG) #math.GR #math.SG #msc:14L30 #msc:14R20 #msc:37K10 #msc:53D17

paper · pdf · doi:10.48550/arxiv.0811.0835

27 pages

arxiv created 2008/11/06 · arxiv updated 2009/12/01

Abstract

In recent work Bertram Kostant and Nolan Wallach ([KW1], [KW2]) have defined an interesting action of a simply connected Lie group A isomorphic to ℂ^{n\choose 2} on gl(n) using a completely integrable system derived from Gelfand-Zeitlin theory. In this paper we show that an analogous action of ℂ^d exists on the complex orthogonal Lie algebra so(n), where d is half the dimension of a regular adjoint orbit in so(n). In [KW1], Kostant and Wallach describe the orbits of A on a certain Zariski open subset of regular semisimple elements in gl(n). We extend these results to the case of so(n). We also make brief mention of the author's results in [Col1], which describe all A-orbits of dimension n\choose 2 in gl(n).

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